src/HOL/Tools/numeral_simprocs.ML
author haftmann
Wed Feb 10 14:12:04 2010 +0100 (2010-02-10)
changeset 35092 cfe605c54e50
parent 35084 e25eedfc15ce
child 35267 8dfd816713c6
permissions -rw-r--r--
moved less_eq, less to Orderings.thy; moved abs, sgn to Groups.thy
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(* Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
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   Copyright   2000  University of Cambridge
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Simprocs for the integer numerals.
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*)
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(*To quote from Provers/Arith/cancel_numeral_factor.ML:
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Cancels common coefficients in balanced expressions:
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     u*#m ~~ u'*#m'  ==  #n*u ~~ #n'*u'
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where ~~ is an appropriate balancing operation (e.g. =, <=, <, div, /)
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and d = gcd(m,m') and n=m/d and n'=m'/d.
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*)
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signature NUMERAL_SIMPROCS =
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sig
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  val assoc_fold_simproc: simproc
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  val combine_numerals: simproc
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  val cancel_numerals: simproc list
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  val cancel_factors: simproc list
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  val cancel_numeral_factors: simproc list
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  val field_combine_numerals: simproc
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  val field_cancel_numeral_factors: simproc list
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  val num_ss: simpset
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end;
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structure Numeral_Simprocs : NUMERAL_SIMPROCS =
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struct
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val mk_number = Arith_Data.mk_number;
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val mk_sum = Arith_Data.mk_sum;
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val long_mk_sum = Arith_Data.long_mk_sum;
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val dest_sum = Arith_Data.dest_sum;
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val mk_diff = HOLogic.mk_binop @{const_name Algebras.minus};
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val dest_diff = HOLogic.dest_bin @{const_name Algebras.minus} Term.dummyT;
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val mk_times = HOLogic.mk_binop @{const_name Algebras.times};
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fun one_of T = Const(@{const_name Algebras.one}, T);
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(* build product with trailing 1 rather than Numeral 1 in order to avoid the
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   unnecessary restriction to type class number_ring
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   which is not required for cancellation of common factors in divisions.
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*)
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fun mk_prod T = 
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  let val one = one_of T
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  fun mk [] = one
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    | mk [t] = t
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    | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
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  in mk end;
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(*This version ALWAYS includes a trailing one*)
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fun long_mk_prod T []        = one_of T
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  | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
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val dest_times = HOLogic.dest_bin @{const_name Algebras.times} Term.dummyT;
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fun dest_prod t =
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      let val (t,u) = dest_times t
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      in dest_prod t @ dest_prod u end
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      handle TERM _ => [t];
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fun find_first_numeral past (t::terms) =
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        ((snd (HOLogic.dest_number t), rev past @ terms)
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         handle TERM _ => find_first_numeral (t::past) terms)
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  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
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(*DON'T do the obvious simplifications; that would create special cases*)
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fun mk_coeff (k, t) = mk_times (mk_number (Term.fastype_of t) k, t);
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(*Express t as a product of (possibly) a numeral with other sorted terms*)
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fun dest_coeff sign (Const (@{const_name Algebras.uminus}, _) $ t) = dest_coeff (~sign) t
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  | dest_coeff sign t =
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    let val ts = sort TermOrd.term_ord (dest_prod t)
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        val (n, ts') = find_first_numeral [] ts
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                          handle TERM _ => (1, ts)
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    in (sign*n, mk_prod (Term.fastype_of t) ts') end;
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(*Find first coefficient-term THAT MATCHES u*)
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fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
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  | find_first_coeff past u (t::terms) =
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        let val (n,u') = dest_coeff 1 t
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        in if u aconv u' then (n, rev past @ terms)
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                         else find_first_coeff (t::past) u terms
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        end
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        handle TERM _ => find_first_coeff (t::past) u terms;
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(*Fractions as pairs of ints. Can't use Rat.rat because the representation
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  needs to preserve negative values in the denominator.*)
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fun mk_frac (p, q) = if q = 0 then raise Div else (p, q);
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(*Don't reduce fractions; sums must be proved by rule add_frac_eq.
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  Fractions are reduced later by the cancel_numeral_factor simproc.*)
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fun add_frac ((p1, q1), (p2, q2)) = (p1 * q2 + p2 * q1, q1 * q2);
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val mk_divide = HOLogic.mk_binop @{const_name Rings.divide};
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(*Build term (p / q) * t*)
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fun mk_fcoeff ((p, q), t) =
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  let val T = Term.fastype_of t
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  in mk_times (mk_divide (mk_number T p, mk_number T q), t) end;
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(*Express t as a product of a fraction with other sorted terms*)
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fun dest_fcoeff sign (Const (@{const_name Algebras.uminus}, _) $ t) = dest_fcoeff (~sign) t
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  | dest_fcoeff sign (Const (@{const_name Rings.divide}, _) $ t $ u) =
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    let val (p, t') = dest_coeff sign t
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        val (q, u') = dest_coeff 1 u
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    in (mk_frac (p, q), mk_divide (t', u')) end
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  | dest_fcoeff sign t =
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    let val (p, t') = dest_coeff sign t
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        val T = Term.fastype_of t
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    in (mk_frac (p, 1), mk_divide (t', one_of T)) end;
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(** New term ordering so that AC-rewriting brings numerals to the front **)
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(*Order integers by absolute value and then by sign. The standard integer
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  ordering is not well-founded.*)
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fun num_ord (i,j) =
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  (case int_ord (abs i, abs j) of
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    EQUAL => int_ord (Int.sign i, Int.sign j) 
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  | ord => ord);
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(*This resembles TermOrd.term_ord, but it puts binary numerals before other
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  non-atomic terms.*)
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local open Term 
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in 
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fun numterm_ord (Abs (_, T, t), Abs(_, U, u)) =
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      (case numterm_ord (t, u) of EQUAL => TermOrd.typ_ord (T, U) | ord => ord)
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  | numterm_ord
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     (Const(@{const_name Int.number_of}, _) $ v, Const(@{const_name Int.number_of}, _) $ w) =
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     num_ord (HOLogic.dest_numeral v, HOLogic.dest_numeral w)
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  | numterm_ord (Const(@{const_name Int.number_of}, _) $ _, _) = LESS
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  | numterm_ord (_, Const(@{const_name Int.number_of}, _) $ _) = GREATER
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  | numterm_ord (t, u) =
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      (case int_ord (size_of_term t, size_of_term u) of
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        EQUAL =>
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          let val (f, ts) = strip_comb t and (g, us) = strip_comb u in
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            (case TermOrd.hd_ord (f, g) of EQUAL => numterms_ord (ts, us) | ord => ord)
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          end
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      | ord => ord)
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and numterms_ord (ts, us) = list_ord numterm_ord (ts, us)
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end;
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fun numtermless tu = (numterm_ord tu = LESS);
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val num_ss = HOL_ss settermless numtermless;
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(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic isn't complicated by the abstract 0 and 1.*)
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val numeral_syms = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym];
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(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1, 1*x, x*1, x/1 *)
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val add_0s =  @{thms add_0s};
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val mult_1s = @{thms mult_1s mult_1_left mult_1_right divide_1};
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(*Simplify inverse Numeral1, a/Numeral1*)
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val inverse_1s = [@{thm inverse_numeral_1}];
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val divide_1s = [@{thm divide_numeral_1}];
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(*To perform binary arithmetic.  The "left" rewriting handles patterns
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  created by the Numeral_Simprocs, such as 3 * (5 * x). *)
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val simps = [@{thm numeral_0_eq_0} RS sym, @{thm numeral_1_eq_1} RS sym,
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                 @{thm add_number_of_left}, @{thm mult_number_of_left}] @
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                @{thms arith_simps} @ @{thms rel_simps};
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(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
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  during re-arrangement*)
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val non_add_simps =
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  subtract Thm.eq_thm [@{thm add_number_of_left}, @{thm number_of_add} RS sym] simps;
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(*To evaluate binary negations of coefficients*)
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val minus_simps = [@{thm numeral_m1_eq_minus_1} RS sym, @{thm number_of_minus} RS sym] @
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                   @{thms minus_bin_simps} @ @{thms pred_bin_simps};
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(*To let us treat subtraction as addition*)
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val diff_simps = [@{thm diff_minus}, @{thm minus_add_distrib}, @{thm minus_minus}];
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(*To let us treat division as multiplication*)
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val divide_simps = [@{thm divide_inverse}, @{thm inverse_mult_distrib}, @{thm inverse_inverse_eq}];
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(*push the unary minus down*)
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val minus_mult_eq_1_to_2 = @{lemma "- (a::'a::ring) * b = a * - b" by simp};
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(*to extract again any uncancelled minuses*)
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val minus_from_mult_simps =
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    [@{thm minus_minus}, @{thm mult_minus_left}, @{thm mult_minus_right}];
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(*combine unary minus with numeric literals, however nested within a product*)
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val mult_minus_simps =
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    [@{thm mult_assoc}, @{thm minus_mult_left}, minus_mult_eq_1_to_2];
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val norm_ss1 = num_ss addsimps numeral_syms @ add_0s @ mult_1s @
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  diff_simps @ minus_simps @ @{thms add_ac}
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val norm_ss2 = num_ss addsimps non_add_simps @ mult_minus_simps
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val norm_ss3 = num_ss addsimps minus_from_mult_simps @ @{thms add_ac} @ @{thms mult_ac}
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structure CancelNumeralsCommon =
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  struct
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  val mk_sum            = mk_sum
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  val dest_sum          = dest_sum
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  val mk_coeff          = mk_coeff
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  val dest_coeff        = dest_coeff 1
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  val find_first_coeff  = find_first_coeff []
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  fun trans_tac _       = Arith_Data.trans_tac
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  fun norm_tac ss =
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    ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
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    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
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  val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
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  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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  val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
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  end;
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structure EqCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = Arith_Data.prove_conv
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  val mk_bal   = HOLogic.mk_eq
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  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
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  val bal_add1 = @{thm eq_add_iff1} RS trans
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  val bal_add2 = @{thm eq_add_iff2} RS trans
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);
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structure LessCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = Arith_Data.prove_conv
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  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
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  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
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  val bal_add1 = @{thm less_add_iff1} RS trans
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  val bal_add2 = @{thm less_add_iff2} RS trans
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);
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structure LeCancelNumerals = CancelNumeralsFun
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 (open CancelNumeralsCommon
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  val prove_conv = Arith_Data.prove_conv
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  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
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  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
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  val bal_add1 = @{thm le_add_iff1} RS trans
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  val bal_add2 = @{thm le_add_iff2} RS trans
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);
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val cancel_numerals =
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  map (Arith_Data.prep_simproc @{theory})
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   [("inteq_cancel_numerals",
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     ["(l::'a::number_ring) + m = n",
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      "(l::'a::number_ring) = m + n",
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      "(l::'a::number_ring) - m = n",
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      "(l::'a::number_ring) = m - n",
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      "(l::'a::number_ring) * m = n",
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      "(l::'a::number_ring) = m * n"],
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     K EqCancelNumerals.proc),
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    ("intless_cancel_numerals",
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     ["(l::'a::{linordered_idom,number_ring}) + m < n",
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      "(l::'a::{linordered_idom,number_ring}) < m + n",
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      "(l::'a::{linordered_idom,number_ring}) - m < n",
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      "(l::'a::{linordered_idom,number_ring}) < m - n",
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      "(l::'a::{linordered_idom,number_ring}) * m < n",
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      "(l::'a::{linordered_idom,number_ring}) < m * n"],
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     K LessCancelNumerals.proc),
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    ("intle_cancel_numerals",
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     ["(l::'a::{linordered_idom,number_ring}) + m <= n",
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      "(l::'a::{linordered_idom,number_ring}) <= m + n",
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      "(l::'a::{linordered_idom,number_ring}) - m <= n",
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      "(l::'a::{linordered_idom,number_ring}) <= m - n",
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      "(l::'a::{linordered_idom,number_ring}) * m <= n",
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      "(l::'a::{linordered_idom,number_ring}) <= m * n"],
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     K LeCancelNumerals.proc)];
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structure CombineNumeralsData =
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  struct
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  type coeff            = int
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  val iszero            = (fn x => x = 0)
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  val add               = op +
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  val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
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  val dest_sum          = dest_sum
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  val mk_coeff          = mk_coeff
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  val dest_coeff        = dest_coeff 1
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  val left_distrib      = @{thm combine_common_factor} RS trans
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  val prove_conv        = Arith_Data.prove_conv_nohyps
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  fun trans_tac _       = Arith_Data.trans_tac
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  fun norm_tac ss =
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    ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
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    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
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    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
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  val numeral_simp_ss = HOL_ss addsimps add_0s @ simps
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  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
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  val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s)
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   295
  end;
haftmann@31068
   296
haftmann@31068
   297
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
haftmann@31068
   298
haftmann@31068
   299
(*Version for fields, where coefficients can be fractions*)
haftmann@31068
   300
structure FieldCombineNumeralsData =
haftmann@31068
   301
  struct
haftmann@31068
   302
  type coeff            = int * int
haftmann@31068
   303
  val iszero            = (fn (p, q) => p = 0)
haftmann@31068
   304
  val add               = add_frac
haftmann@31068
   305
  val mk_sum            = long_mk_sum
haftmann@31068
   306
  val dest_sum          = dest_sum
haftmann@31068
   307
  val mk_coeff          = mk_fcoeff
haftmann@31068
   308
  val dest_coeff        = dest_fcoeff 1
haftmann@31068
   309
  val left_distrib      = @{thm combine_common_factor} RS trans
haftmann@31068
   310
  val prove_conv        = Arith_Data.prove_conv_nohyps
wenzelm@31368
   311
  fun trans_tac _       = Arith_Data.trans_tac
haftmann@31068
   312
haftmann@31068
   313
  val norm_ss1a = norm_ss1 addsimps inverse_1s @ divide_simps
haftmann@31068
   314
  fun norm_tac ss =
haftmann@31068
   315
    ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1a))
haftmann@31068
   316
    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
haftmann@31068
   317
    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
haftmann@31068
   318
haftmann@31068
   319
  val numeral_simp_ss = HOL_ss addsimps add_0s @ simps @ [@{thm add_frac_eq}]
haftmann@31068
   320
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
haftmann@31068
   321
  val simplify_meta_eq = Arith_Data.simplify_meta_eq (add_0s @ mult_1s @ divide_1s)
haftmann@31068
   322
  end;
haftmann@31068
   323
haftmann@31068
   324
structure FieldCombineNumerals = CombineNumeralsFun(FieldCombineNumeralsData);
haftmann@31068
   325
haftmann@31068
   326
val combine_numerals =
wenzelm@32155
   327
  Arith_Data.prep_simproc @{theory}
haftmann@31068
   328
    ("int_combine_numerals", 
haftmann@31068
   329
     ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
haftmann@31068
   330
     K CombineNumerals.proc);
haftmann@31068
   331
haftmann@31068
   332
val field_combine_numerals =
wenzelm@32155
   333
  Arith_Data.prep_simproc @{theory}
haftmann@31068
   334
    ("field_combine_numerals", 
haftmann@31068
   335
     ["(i::'a::{number_ring,field,division_by_zero}) + j",
haftmann@31068
   336
      "(i::'a::{number_ring,field,division_by_zero}) - j"], 
haftmann@31068
   337
     K FieldCombineNumerals.proc);
haftmann@31068
   338
haftmann@31068
   339
(** Constant folding for multiplication in semirings **)
haftmann@31068
   340
haftmann@31068
   341
(*We do not need folding for addition: combine_numerals does the same thing*)
haftmann@31068
   342
haftmann@31068
   343
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
haftmann@31068
   344
struct
haftmann@31068
   345
  val assoc_ss = HOL_ss addsimps @{thms mult_ac}
haftmann@31068
   346
  val eq_reflection = eq_reflection
haftmann@31068
   347
  fun is_numeral (Const(@{const_name Int.number_of}, _) $ _) = true
haftmann@31068
   348
    | is_numeral _ = false;
haftmann@31068
   349
end;
haftmann@31068
   350
haftmann@31068
   351
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
haftmann@31068
   352
haftmann@31068
   353
val assoc_fold_simproc =
wenzelm@32155
   354
  Arith_Data.prep_simproc @{theory}
haftmann@31068
   355
   ("semiring_assoc_fold", ["(a::'a::comm_semiring_1_cancel) * b"],
haftmann@31068
   356
    K Semiring_Times_Assoc.proc);
wenzelm@23164
   357
wenzelm@23164
   358
structure CancelNumeralFactorCommon =
wenzelm@23164
   359
  struct
wenzelm@23164
   360
  val mk_coeff          = mk_coeff
wenzelm@23164
   361
  val dest_coeff        = dest_coeff 1
wenzelm@31368
   362
  fun trans_tac _       = Arith_Data.trans_tac
wenzelm@23164
   363
wenzelm@23164
   364
  val norm_ss1 = HOL_ss addsimps minus_from_mult_simps @ mult_1s
wenzelm@23164
   365
  val norm_ss2 = HOL_ss addsimps simps @ mult_minus_simps
haftmann@23881
   366
  val norm_ss3 = HOL_ss addsimps @{thms mult_ac}
wenzelm@23164
   367
  fun norm_tac ss =
wenzelm@23164
   368
    ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss1))
wenzelm@23164
   369
    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss2))
wenzelm@23164
   370
    THEN ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss3))
wenzelm@23164
   371
haftmann@31068
   372
  val numeral_simp_ss = HOL_ss addsimps
haftmann@31068
   373
    [@{thm eq_number_of_eq}, @{thm less_number_of}, @{thm le_number_of}] @ simps
wenzelm@23164
   374
  fun numeral_simp_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss numeral_simp_ss))
haftmann@30518
   375
  val simplify_meta_eq = Arith_Data.simplify_meta_eq
haftmann@35064
   376
    [@{thm Nat.add_0}, @{thm Nat.add_0_right}, @{thm mult_zero_left},
huffman@26086
   377
      @{thm mult_zero_right}, @{thm mult_Bit1}, @{thm mult_1_right}];
wenzelm@23164
   378
  end
wenzelm@23164
   379
haftmann@30931
   380
(*Version for semiring_div*)
haftmann@30931
   381
structure DivCancelNumeralFactor = CancelNumeralFactorFun
wenzelm@23164
   382
 (open CancelNumeralFactorCommon
haftmann@30496
   383
  val prove_conv = Arith_Data.prove_conv
wenzelm@23164
   384
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
haftmann@30931
   385
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
haftmann@30931
   386
  val cancel = @{thm div_mult_mult1} RS trans
wenzelm@23164
   387
  val neg_exchanges = false
wenzelm@23164
   388
)
wenzelm@23164
   389
wenzelm@23164
   390
(*Version for fields*)
wenzelm@23164
   391
structure DivideCancelNumeralFactor = CancelNumeralFactorFun
wenzelm@23164
   392
 (open CancelNumeralFactorCommon
haftmann@30496
   393
  val prove_conv = Arith_Data.prove_conv
haftmann@35084
   394
  val mk_bal   = HOLogic.mk_binop @{const_name Rings.divide}
haftmann@35084
   395
  val dest_bal = HOLogic.dest_bin @{const_name Rings.divide} Term.dummyT
nipkow@23413
   396
  val cancel = @{thm mult_divide_mult_cancel_left} RS trans
wenzelm@23164
   397
  val neg_exchanges = false
wenzelm@23164
   398
)
wenzelm@23164
   399
wenzelm@23164
   400
structure EqCancelNumeralFactor = CancelNumeralFactorFun
wenzelm@23164
   401
 (open CancelNumeralFactorCommon
haftmann@30496
   402
  val prove_conv = Arith_Data.prove_conv
wenzelm@23164
   403
  val mk_bal   = HOLogic.mk_eq
wenzelm@23164
   404
  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
wenzelm@23164
   405
  val cancel = @{thm mult_cancel_left} RS trans
wenzelm@23164
   406
  val neg_exchanges = false
wenzelm@23164
   407
)
wenzelm@23164
   408
wenzelm@23164
   409
structure LessCancelNumeralFactor = CancelNumeralFactorFun
wenzelm@23164
   410
 (open CancelNumeralFactorCommon
haftmann@30496
   411
  val prove_conv = Arith_Data.prove_conv
haftmann@35092
   412
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
haftmann@35092
   413
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
wenzelm@23164
   414
  val cancel = @{thm mult_less_cancel_left} RS trans
wenzelm@23164
   415
  val neg_exchanges = true
wenzelm@23164
   416
)
wenzelm@23164
   417
wenzelm@23164
   418
structure LeCancelNumeralFactor = CancelNumeralFactorFun
wenzelm@23164
   419
 (open CancelNumeralFactorCommon
haftmann@30496
   420
  val prove_conv = Arith_Data.prove_conv
haftmann@35092
   421
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
haftmann@35092
   422
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
wenzelm@23164
   423
  val cancel = @{thm mult_le_cancel_left} RS trans
wenzelm@23164
   424
  val neg_exchanges = true
wenzelm@23164
   425
)
wenzelm@23164
   426
wenzelm@23164
   427
val cancel_numeral_factors =
wenzelm@32155
   428
  map (Arith_Data.prep_simproc @{theory})
wenzelm@23164
   429
   [("ring_eq_cancel_numeral_factor",
wenzelm@23164
   430
     ["(l::'a::{idom,number_ring}) * m = n",
wenzelm@23164
   431
      "(l::'a::{idom,number_ring}) = m * n"],
wenzelm@23164
   432
     K EqCancelNumeralFactor.proc),
wenzelm@23164
   433
    ("ring_less_cancel_numeral_factor",
haftmann@35028
   434
     ["(l::'a::{linordered_idom,number_ring}) * m < n",
haftmann@35028
   435
      "(l::'a::{linordered_idom,number_ring}) < m * n"],
wenzelm@23164
   436
     K LessCancelNumeralFactor.proc),
wenzelm@23164
   437
    ("ring_le_cancel_numeral_factor",
haftmann@35028
   438
     ["(l::'a::{linordered_idom,number_ring}) * m <= n",
haftmann@35028
   439
      "(l::'a::{linordered_idom,number_ring}) <= m * n"],
wenzelm@23164
   440
     K LeCancelNumeralFactor.proc),
wenzelm@23164
   441
    ("int_div_cancel_numeral_factors",
haftmann@30931
   442
     ["((l::'a::{semiring_div,number_ring}) * m) div n",
haftmann@30931
   443
      "(l::'a::{semiring_div,number_ring}) div (m * n)"],
haftmann@30931
   444
     K DivCancelNumeralFactor.proc),
wenzelm@23164
   445
    ("divide_cancel_numeral_factor",
wenzelm@23164
   446
     ["((l::'a::{division_by_zero,field,number_ring}) * m) / n",
wenzelm@23164
   447
      "(l::'a::{division_by_zero,field,number_ring}) / (m * n)",
wenzelm@23164
   448
      "((number_of v)::'a::{division_by_zero,field,number_ring}) / (number_of w)"],
wenzelm@23164
   449
     K DivideCancelNumeralFactor.proc)];
wenzelm@23164
   450
wenzelm@23164
   451
val field_cancel_numeral_factors =
wenzelm@32155
   452
  map (Arith_Data.prep_simproc @{theory})
wenzelm@23164
   453
   [("field_eq_cancel_numeral_factor",
wenzelm@23164
   454
     ["(l::'a::{field,number_ring}) * m = n",
wenzelm@23164
   455
      "(l::'a::{field,number_ring}) = m * n"],
wenzelm@23164
   456
     K EqCancelNumeralFactor.proc),
wenzelm@23164
   457
    ("field_cancel_numeral_factor",
wenzelm@23164
   458
     ["((l::'a::{division_by_zero,field,number_ring}) * m) / n",
wenzelm@23164
   459
      "(l::'a::{division_by_zero,field,number_ring}) / (m * n)",
wenzelm@23164
   460
      "((number_of v)::'a::{division_by_zero,field,number_ring}) / (number_of w)"],
wenzelm@23164
   461
     K DivideCancelNumeralFactor.proc)]
wenzelm@23164
   462
wenzelm@23164
   463
wenzelm@23164
   464
(** Declarations for ExtractCommonTerm **)
wenzelm@23164
   465
wenzelm@23164
   466
(*Find first term that matches u*)
wenzelm@23164
   467
fun find_first_t past u []         = raise TERM ("find_first_t", [])
wenzelm@23164
   468
  | find_first_t past u (t::terms) =
wenzelm@23164
   469
        if u aconv t then (rev past @ terms)
wenzelm@23164
   470
        else find_first_t (t::past) u terms
wenzelm@23164
   471
        handle TERM _ => find_first_t (t::past) u terms;
wenzelm@23164
   472
wenzelm@23164
   473
(** Final simplification for the CancelFactor simprocs **)
haftmann@30518
   474
val simplify_one = Arith_Data.simplify_meta_eq  
nipkow@30031
   475
  [@{thm mult_1_left}, @{thm mult_1_right}, @{thm div_by_1}, @{thm numeral_1_eq_1}];
wenzelm@23164
   476
nipkow@30649
   477
fun cancel_simplify_meta_eq ss cancel_th th =
wenzelm@23164
   478
    simplify_one ss (([th, cancel_th]) MRS trans);
wenzelm@23164
   479
nipkow@30649
   480
local
haftmann@31067
   481
  val Tp_Eq = Thm.reflexive (Thm.cterm_of @{theory HOL} HOLogic.Trueprop)
nipkow@30649
   482
  fun Eq_True_elim Eq = 
nipkow@30649
   483
    Thm.equal_elim (Thm.combination Tp_Eq (Thm.symmetric Eq)) @{thm TrueI}
nipkow@30649
   484
in
nipkow@30649
   485
fun sign_conv pos_th neg_th ss t =
nipkow@30649
   486
  let val T = fastype_of t;
haftmann@34974
   487
      val zero = Const(@{const_name Algebras.zero}, T);
haftmann@35092
   488
      val less = Const(@{const_name Orderings.less}, [T,T] ---> HOLogic.boolT);
nipkow@30649
   489
      val pos = less $ zero $ t and neg = less $ t $ zero
nipkow@30649
   490
      fun prove p =
haftmann@31101
   491
        Option.map Eq_True_elim (Lin_Arith.simproc ss p)
nipkow@30649
   492
        handle THM _ => NONE
nipkow@30649
   493
    in case prove pos of
nipkow@30649
   494
         SOME th => SOME(th RS pos_th)
nipkow@30649
   495
       | NONE => (case prove neg of
nipkow@30649
   496
                    SOME th => SOME(th RS neg_th)
nipkow@30649
   497
                  | NONE => NONE)
nipkow@30649
   498
    end;
nipkow@30649
   499
end
nipkow@30649
   500
wenzelm@23164
   501
structure CancelFactorCommon =
wenzelm@23164
   502
  struct
wenzelm@23164
   503
  val mk_sum            = long_mk_prod
wenzelm@23164
   504
  val dest_sum          = dest_prod
wenzelm@23164
   505
  val mk_coeff          = mk_coeff
wenzelm@23164
   506
  val dest_coeff        = dest_coeff
wenzelm@23164
   507
  val find_first        = find_first_t []
wenzelm@31368
   508
  fun trans_tac _       = Arith_Data.trans_tac
haftmann@23881
   509
  val norm_ss = HOL_ss addsimps mult_1s @ @{thms mult_ac}
wenzelm@23164
   510
  fun norm_tac ss = ALLGOALS (simp_tac (Simplifier.inherit_context ss norm_ss))
nipkow@30649
   511
  val simplify_meta_eq  = cancel_simplify_meta_eq 
wenzelm@23164
   512
  end;
wenzelm@23164
   513
wenzelm@23164
   514
(*mult_cancel_left requires a ring with no zero divisors.*)
wenzelm@23164
   515
structure EqCancelFactor = ExtractCommonTermFun
wenzelm@23164
   516
 (open CancelFactorCommon
haftmann@30496
   517
  val prove_conv = Arith_Data.prove_conv
wenzelm@23164
   518
  val mk_bal   = HOLogic.mk_eq
wenzelm@23164
   519
  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
wenzelm@31368
   520
  fun simp_conv _ _ = SOME @{thm mult_cancel_left}
nipkow@30649
   521
);
nipkow@30649
   522
nipkow@30649
   523
(*for ordered rings*)
nipkow@30649
   524
structure LeCancelFactor = ExtractCommonTermFun
nipkow@30649
   525
 (open CancelFactorCommon
nipkow@30649
   526
  val prove_conv = Arith_Data.prove_conv
haftmann@35092
   527
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less_eq}
haftmann@35092
   528
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less_eq} Term.dummyT
nipkow@30649
   529
  val simp_conv = sign_conv
nipkow@30649
   530
    @{thm mult_le_cancel_left_pos} @{thm mult_le_cancel_left_neg}
nipkow@30649
   531
);
nipkow@30649
   532
nipkow@30649
   533
(*for ordered rings*)
nipkow@30649
   534
structure LessCancelFactor = ExtractCommonTermFun
nipkow@30649
   535
 (open CancelFactorCommon
nipkow@30649
   536
  val prove_conv = Arith_Data.prove_conv
haftmann@35092
   537
  val mk_bal   = HOLogic.mk_binrel @{const_name Orderings.less}
haftmann@35092
   538
  val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} Term.dummyT
nipkow@30649
   539
  val simp_conv = sign_conv
nipkow@30649
   540
    @{thm mult_less_cancel_left_pos} @{thm mult_less_cancel_left_neg}
wenzelm@23164
   541
);
wenzelm@23164
   542
haftmann@30931
   543
(*for semirings with division*)
haftmann@30931
   544
structure DivCancelFactor = ExtractCommonTermFun
wenzelm@23164
   545
 (open CancelFactorCommon
haftmann@30496
   546
  val prove_conv = Arith_Data.prove_conv
wenzelm@23164
   547
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.div}
haftmann@30931
   548
  val dest_bal = HOLogic.dest_bin @{const_name Divides.div} Term.dummyT
wenzelm@31368
   549
  fun simp_conv _ _ = SOME @{thm div_mult_mult1_if}
wenzelm@23164
   550
);
wenzelm@23164
   551
haftmann@30931
   552
structure ModCancelFactor = ExtractCommonTermFun
nipkow@24395
   553
 (open CancelFactorCommon
haftmann@30496
   554
  val prove_conv = Arith_Data.prove_conv
nipkow@24395
   555
  val mk_bal   = HOLogic.mk_binop @{const_name Divides.mod}
haftmann@31067
   556
  val dest_bal = HOLogic.dest_bin @{const_name Divides.mod} Term.dummyT
wenzelm@31368
   557
  fun simp_conv _ _ = SOME @{thm mod_mult_mult1}
nipkow@24395
   558
);
nipkow@24395
   559
haftmann@30931
   560
(*for idoms*)
haftmann@30931
   561
structure DvdCancelFactor = ExtractCommonTermFun
nipkow@23969
   562
 (open CancelFactorCommon
haftmann@30496
   563
  val prove_conv = Arith_Data.prove_conv
haftmann@35050
   564
  val mk_bal   = HOLogic.mk_binrel @{const_name Rings.dvd}
haftmann@35050
   565
  val dest_bal = HOLogic.dest_bin @{const_name Rings.dvd} Term.dummyT
wenzelm@31368
   566
  fun simp_conv _ _ = SOME @{thm dvd_mult_cancel_left}
nipkow@23969
   567
);
nipkow@23969
   568
wenzelm@23164
   569
(*Version for all fields, including unordered ones (type complex).*)
wenzelm@23164
   570
structure DivideCancelFactor = ExtractCommonTermFun
wenzelm@23164
   571
 (open CancelFactorCommon
haftmann@30496
   572
  val prove_conv = Arith_Data.prove_conv
haftmann@35084
   573
  val mk_bal   = HOLogic.mk_binop @{const_name Rings.divide}
haftmann@35084
   574
  val dest_bal = HOLogic.dest_bin @{const_name Rings.divide} Term.dummyT
wenzelm@31368
   575
  fun simp_conv _ _ = SOME @{thm mult_divide_mult_cancel_left_if}
wenzelm@23164
   576
);
wenzelm@23164
   577
wenzelm@23164
   578
val cancel_factors =
wenzelm@32155
   579
  map (Arith_Data.prep_simproc @{theory})
wenzelm@23164
   580
   [("ring_eq_cancel_factor",
haftmann@30931
   581
     ["(l::'a::idom) * m = n",
haftmann@30931
   582
      "(l::'a::idom) = m * n"],
nipkow@30649
   583
     K EqCancelFactor.proc),
haftmann@35043
   584
    ("linordered_ring_le_cancel_factor",
haftmann@35028
   585
     ["(l::'a::linordered_ring) * m <= n",
haftmann@35028
   586
      "(l::'a::linordered_ring) <= m * n"],
nipkow@30649
   587
     K LeCancelFactor.proc),
haftmann@35043
   588
    ("linordered_ring_less_cancel_factor",
haftmann@35028
   589
     ["(l::'a::linordered_ring) * m < n",
haftmann@35028
   590
      "(l::'a::linordered_ring) < m * n"],
nipkow@30649
   591
     K LessCancelFactor.proc),
wenzelm@23164
   592
    ("int_div_cancel_factor",
haftmann@30931
   593
     ["((l::'a::semiring_div) * m) div n", "(l::'a::semiring_div) div (m * n)"],
haftmann@30931
   594
     K DivCancelFactor.proc),
nipkow@24395
   595
    ("int_mod_cancel_factor",
haftmann@30931
   596
     ["((l::'a::semiring_div) * m) mod n", "(l::'a::semiring_div) mod (m * n)"],
haftmann@30931
   597
     K ModCancelFactor.proc),
huffman@29981
   598
    ("dvd_cancel_factor",
huffman@29981
   599
     ["((l::'a::idom) * m) dvd n", "(l::'a::idom) dvd (m * n)"],
haftmann@30931
   600
     K DvdCancelFactor.proc),
wenzelm@23164
   601
    ("divide_cancel_factor",
nipkow@23400
   602
     ["((l::'a::{division_by_zero,field}) * m) / n",
nipkow@23400
   603
      "(l::'a::{division_by_zero,field}) / (m * n)"],
wenzelm@23164
   604
     K DivideCancelFactor.proc)];
wenzelm@23164
   605
wenzelm@23164
   606
end;
wenzelm@23164
   607
haftmann@31068
   608
Addsimprocs Numeral_Simprocs.cancel_numerals;
haftmann@31068
   609
Addsimprocs [Numeral_Simprocs.combine_numerals];
haftmann@31068
   610
Addsimprocs [Numeral_Simprocs.field_combine_numerals];
haftmann@31068
   611
Addsimprocs [Numeral_Simprocs.assoc_fold_simproc];
haftmann@31068
   612
haftmann@31068
   613
(*examples:
haftmann@31068
   614
print_depth 22;
haftmann@31068
   615
set timing;
haftmann@31068
   616
set trace_simp;
haftmann@31068
   617
fun test s = (Goal s, by (Simp_tac 1));
haftmann@31068
   618
haftmann@31068
   619
test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
haftmann@31068
   620
haftmann@31068
   621
test "2*u = (u::int)";
haftmann@31068
   622
test "(i + j + 12 + (k::int)) - 15 = y";
haftmann@31068
   623
test "(i + j + 12 + (k::int)) - 5 = y";
haftmann@31068
   624
haftmann@31068
   625
test "y - b < (b::int)";
haftmann@31068
   626
test "y - (3*b + c) < (b::int) - 2*c";
haftmann@31068
   627
haftmann@31068
   628
test "(2*x - (u*v) + y) - v*3*u = (w::int)";
haftmann@31068
   629
test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
haftmann@31068
   630
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
haftmann@31068
   631
test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
haftmann@31068
   632
haftmann@31068
   633
test "(i + j + 12 + (k::int)) = u + 15 + y";
haftmann@31068
   634
test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
haftmann@31068
   635
haftmann@31068
   636
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";
haftmann@31068
   637
haftmann@31068
   638
test "a + -(b+c) + b = (d::int)";
haftmann@31068
   639
test "a + -(b+c) - b = (d::int)";
haftmann@31068
   640
haftmann@31068
   641
(*negative numerals*)
haftmann@31068
   642
test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
haftmann@31068
   643
test "(i + j + -3 + (k::int)) < u + 5 + y";
haftmann@31068
   644
test "(i + j + 3 + (k::int)) < u + -6 + y";
haftmann@31068
   645
test "(i + j + -12 + (k::int)) - 15 = y";
haftmann@31068
   646
test "(i + j + 12 + (k::int)) - -15 = y";
haftmann@31068
   647
test "(i + j + -12 + (k::int)) - -15 = y";
haftmann@31068
   648
*)
haftmann@31068
   649
haftmann@31068
   650
Addsimprocs Numeral_Simprocs.cancel_numeral_factors;
haftmann@31068
   651
haftmann@31068
   652
(*examples:
haftmann@31068
   653
print_depth 22;
haftmann@31068
   654
set timing;
haftmann@31068
   655
set trace_simp;
haftmann@31068
   656
fun test s = (Goal s; by (Simp_tac 1));
haftmann@31068
   657
haftmann@31068
   658
test "9*x = 12 * (y::int)";
haftmann@31068
   659
test "(9*x) div (12 * (y::int)) = z";
haftmann@31068
   660
test "9*x < 12 * (y::int)";
haftmann@31068
   661
test "9*x <= 12 * (y::int)";
haftmann@31068
   662
haftmann@31068
   663
test "-99*x = 132 * (y::int)";
haftmann@31068
   664
test "(-99*x) div (132 * (y::int)) = z";
haftmann@31068
   665
test "-99*x < 132 * (y::int)";
haftmann@31068
   666
test "-99*x <= 132 * (y::int)";
haftmann@31068
   667
haftmann@31068
   668
test "999*x = -396 * (y::int)";
haftmann@31068
   669
test "(999*x) div (-396 * (y::int)) = z";
haftmann@31068
   670
test "999*x < -396 * (y::int)";
haftmann@31068
   671
test "999*x <= -396 * (y::int)";
haftmann@31068
   672
haftmann@31068
   673
test "-99*x = -81 * (y::int)";
haftmann@31068
   674
test "(-99*x) div (-81 * (y::int)) = z";
haftmann@31068
   675
test "-99*x <= -81 * (y::int)";
haftmann@31068
   676
test "-99*x < -81 * (y::int)";
haftmann@31068
   677
haftmann@31068
   678
test "-2 * x = -1 * (y::int)";
haftmann@31068
   679
test "-2 * x = -(y::int)";
haftmann@31068
   680
test "(-2 * x) div (-1 * (y::int)) = z";
haftmann@31068
   681
test "-2 * x < -(y::int)";
haftmann@31068
   682
test "-2 * x <= -1 * (y::int)";
haftmann@31068
   683
test "-x < -23 * (y::int)";
haftmann@31068
   684
test "-x <= -23 * (y::int)";
haftmann@31068
   685
*)
haftmann@31068
   686
haftmann@31068
   687
(*And the same examples for fields such as rat or real:
haftmann@31068
   688
test "0 <= (y::rat) * -2";
haftmann@31068
   689
test "9*x = 12 * (y::rat)";
haftmann@31068
   690
test "(9*x) / (12 * (y::rat)) = z";
haftmann@31068
   691
test "9*x < 12 * (y::rat)";
haftmann@31068
   692
test "9*x <= 12 * (y::rat)";
haftmann@31068
   693
haftmann@31068
   694
test "-99*x = 132 * (y::rat)";
haftmann@31068
   695
test "(-99*x) / (132 * (y::rat)) = z";
haftmann@31068
   696
test "-99*x < 132 * (y::rat)";
haftmann@31068
   697
test "-99*x <= 132 * (y::rat)";
haftmann@31068
   698
haftmann@31068
   699
test "999*x = -396 * (y::rat)";
haftmann@31068
   700
test "(999*x) / (-396 * (y::rat)) = z";
haftmann@31068
   701
test "999*x < -396 * (y::rat)";
haftmann@31068
   702
test "999*x <= -396 * (y::rat)";
haftmann@31068
   703
haftmann@31068
   704
test  "(- ((2::rat) * x) <= 2 * y)";
haftmann@31068
   705
test "-99*x = -81 * (y::rat)";
haftmann@31068
   706
test "(-99*x) / (-81 * (y::rat)) = z";
haftmann@31068
   707
test "-99*x <= -81 * (y::rat)";
haftmann@31068
   708
test "-99*x < -81 * (y::rat)";
haftmann@31068
   709
haftmann@31068
   710
test "-2 * x = -1 * (y::rat)";
haftmann@31068
   711
test "-2 * x = -(y::rat)";
haftmann@31068
   712
test "(-2 * x) / (-1 * (y::rat)) = z";
haftmann@31068
   713
test "-2 * x < -(y::rat)";
haftmann@31068
   714
test "-2 * x <= -1 * (y::rat)";
haftmann@31068
   715
test "-x < -23 * (y::rat)";
haftmann@31068
   716
test "-x <= -23 * (y::rat)";
haftmann@31068
   717
*)
haftmann@31068
   718
haftmann@31068
   719
Addsimprocs Numeral_Simprocs.cancel_factors;
wenzelm@23164
   720
wenzelm@23164
   721
wenzelm@23164
   722
(*examples:
wenzelm@23164
   723
print_depth 22;
wenzelm@23164
   724
set timing;
wenzelm@23164
   725
set trace_simp;
wenzelm@23164
   726
fun test s = (Goal s; by (Asm_simp_tac 1));
wenzelm@23164
   727
wenzelm@23164
   728
test "x*k = k*(y::int)";
wenzelm@23164
   729
test "k = k*(y::int)";
wenzelm@23164
   730
test "a*(b*c) = (b::int)";
wenzelm@23164
   731
test "a*(b*c) = d*(b::int)*(x*a)";
wenzelm@23164
   732
wenzelm@23164
   733
test "(x*k) div (k*(y::int)) = (uu::int)";
wenzelm@23164
   734
test "(k) div (k*(y::int)) = (uu::int)";
wenzelm@23164
   735
test "(a*(b*c)) div ((b::int)) = (uu::int)";
wenzelm@23164
   736
test "(a*(b*c)) div (d*(b::int)*(x*a)) = (uu::int)";
wenzelm@23164
   737
*)
wenzelm@23164
   738
wenzelm@23164
   739
(*And the same examples for fields such as rat or real:
wenzelm@23164
   740
print_depth 22;
wenzelm@23164
   741
set timing;
wenzelm@23164
   742
set trace_simp;
wenzelm@23164
   743
fun test s = (Goal s; by (Asm_simp_tac 1));
wenzelm@23164
   744
wenzelm@23164
   745
test "x*k = k*(y::rat)";
wenzelm@23164
   746
test "k = k*(y::rat)";
wenzelm@23164
   747
test "a*(b*c) = (b::rat)";
wenzelm@23164
   748
test "a*(b*c) = d*(b::rat)*(x*a)";
wenzelm@23164
   749
wenzelm@23164
   750
wenzelm@23164
   751
test "(x*k) / (k*(y::rat)) = (uu::rat)";
wenzelm@23164
   752
test "(k) / (k*(y::rat)) = (uu::rat)";
wenzelm@23164
   753
test "(a*(b*c)) / ((b::rat)) = (uu::rat)";
wenzelm@23164
   754
test "(a*(b*c)) / (d*(b::rat)*(x*a)) = (uu::rat)";
wenzelm@23164
   755
wenzelm@23164
   756
(*FIXME: what do we do about this?*)
wenzelm@23164
   757
test "a*(b*c)/(y*z) = d*(b::rat)*(x*a)/z";
wenzelm@23164
   758
*)